PHYSICAL REVIEW A 84, 033844 (2011)

Optimized heralding schemes for single photons

Yu-Ping Huang, Joseph B. Altepeter, and Prem KumarCenter for Photonic Communication and Computing, EECS Department, Northwestern University,

2145 Sheridan Road, Evanston, Illinois 60208-3118, USA(Received 1 June 2011; published 20 September 2011)

A major obstacle to a practical, heralded source of single photons is the fundamental trade-off between highpurity and high production rate. To overcome this difficulty, we propose applying sequential spectral and temporalfiltering on the signal photons before they are detected for heralding. Based on a multimode theory that takesinto account the effect of simultaneous multiple photon-pair emission, we find that these filters can be optimizedto yield both high purity and a high production rate. While the optimization conditions vary depending on theunderlying photon-pair spectral correlations, all correlation profiles can lead to similarly high performance levelswhen optimized filters are employed. This suggests that a better strategy for improving the performance ofheralded single-photon sources is to adopt an appropriate measurement scheme for the signal photons, ratherthan tailoring the properties of the photon-pair generation medium.

DOI: 10.1103/PhysRevA.84.033844 PACS number(s): 42.50.Dv, 03.67.−a, 03.65.Ta

I. INTRODUCTION

The performance of many single-photon-based applications[1], such as optical quantum computing [2], quantum cryp-tography [3], and long-distance entanglement distribution [4],critically depends on the ability to create single photons in purespatiotemporal states and at high rates. There exist two basictypes of single-photon sources to fulfill this need. The first isbased on antibunched emission from single-emitter systems,such as atoms [5], ions [6], molecules [7], semiconductors [8],and quantum dots [9]. The second is based on “heralding,”in which one photon (the “signal”) is detected and usedas a trigger to herald the presence of a paired photon (the“idler”). Sources of this type have been demonstrated mainlyin guided nonlinear media, such as χ (2) waveguides [10–15]and χ (3) fibers [16–19]. Heralded sources using guided mediahave one key advantage: The photons are generated in asingle, well-defined spatial mode, as opposed to a mixture ofmodes as in the case of single-emitter sources. Single photonsproduced in monomode fibers via heralding, in particular, canbe losslessly coupled into standard telecom fibers [20]. Theythus have the potential to be a valuable resource for networkedquantum information processing capable of harnessing theexisting fiber-based telecommunications infrastructure.

Although heralded single-photon sources have severaladvantages, it is challenging to achieve simultaneously highpurity and a high production rate, as such sources areconstrained by a fundamental trade-off between the two. Forexample, a common method for achieving high purity is toapply narrowband spectral filtering to the signal photons beforethey are detected [20–24]. This method rejects most of theusable pairs, resulting in a low single-photon production rate.An alternate method is to use spectrally factorable photon-pairstates [14,15,18,25–29]. In such a method, however, one has towait for a relatively long time before a second short-durationphoton can be heralded. As a result, the repetition rate ofthe heralded single-photon creation is restricted, resulting in arelatively low production rate.

To overcome these difficulties, we have recently proposedan approach to heralding pure single photons which doesnot rely on narrowband filtering or factorable photon-pair

states [30]. The idea is to apply appropriate spectral andtemporal filtering to the signal photons, so that when detected,they collapse onto a single spectral (temporal) state. Basedon a simplified model, neglecting the effect of simultaneousmultiple photon-pair emission, we have demonstrated viasimulation that both high purity and a high production ratecan be simultaneously achieved, regardless of the spectralcorrelation properties of the paired photons.

In this paper, we extend our previous study [30] to includethe effect of multiple photon-pair emission. Our goal is toidentify the appropriate spectral and temporal filters for the var-ious photon-pair correlations, for which the trade-off betweenhigh purity and high production rate is optimally mitigated.To this end, we develop a multimode theory which describesthe heralded creation of single photons in the presence ofmultipair emission. We then use the theory to numericallyoptimize the spectral and temporal filter widths for a varietyof photon-pair correlations. The results reveal that whilethe optimization conditions vary for different correlations,the final performance using optimized filters is always quitesimilar. This interesting phenomenon suggests that the key toachieving high performance in the heralded creation of singlephotons is to apply an appropriate measurement scheme tothe signal photons. This approach is distinctly superior totailoring the phase-matching properties of nonlinear mediain order to obtain spectrally factorable photon-pair states[14,15,18,25–29].

The paper is organized as follows: We present the multi-mode heralding theory in Sec. II, study the optimization ofmeasurement schemes in Sec. III, and then conclude briefly inSec. IV.

II. MULTIMODE THEORY OF HERALDINGSINGLE PHOTONS

In this section, we develop a multimode theory modelingthe heralded creation of single photons that generally appliesto χ (2) waveguides and χ (3) fibers [31]. In this model, photonpairs are created via spontaneous parametric down-conversion(SPDC) or spontaneous four-wave mixing (SFWM) driven bypulsed pumps. The signal photons are passed through spectral

033844-11050-2947/2011/84(3)/033844(7) ©2011 American Physical Society

YU-PING HUANG, JOSEPH B. ALTEPETER, AND PREM KUMAR PHYSICAL REVIEW A 84, 033844 (2011)

and temporal filters before being measured by an on-offdetector [32]. A “click” of the detector heralds the presenceof at least one idler photon. For the sake of computationalsimplicity, we assume a rectangular-shaped spectral filter withan angular-frequency bandwidth 2πB, where B is in hertz.For the temporal filter, we consider a rectangular shutter witha T -second time window that is shorter than the inherent timeresolution of the detector. Such a time shutter can be builtusing, for example, an ultrafast all-optical quantum switch[33,34].

We first compute the output photon-pair states from awaveguide or a fiber pumped by light pulses. Linearizedaround perfect phase-matching frequencies [26], the effectiveHamiltonian describing photon-pair generation via SPDC orSFWM is given in the undepleted-pump approximation by[35,36]

H (z) = hκ

∫dνsdνiφ(νs + νi)e

i(βsνs+βiνi )z

× a†s (νs)a

†i (νi) + H.c. (1)

Here, κ is proportional to the pair-generation efficiency; 2πνs

and 2πνi are the angular-frequency detunings of the signaland idler photons, respectively, from the phase-matchingfrequencies; and the sum-frequency profile φ(νs + νi) isdetermined by the pump spectrum. For this paper, we consider

φ(νs + νi) = e−(νs+νi )2/2σ 2, (2)

where 2σ is the profile bandwidth in hertz, which equals thefull width at e−1 (e−2) maximum of the pump spectrum inthe case of SPDC (SFWM). In Eq. (1), βs,i are related tothe group-velocity dispersion of the signal and idler photons,respectively; and a

†s (νs) and a

†i (νi) are the creation operators,

respectively, for the signal and idler photons, satisfying[ap(νp),a†

q(ν ′q)] = δp,qδ(νp − ν ′

q) for p,q = s,i. With thisHamiltonian, the quantum state of the output photon pairsfrom a source of length L can be obtained perturbatively via

|〉 = |vac〉 − i

h

∫ L/2

−L/2dzH (z)|vac〉

− 1

h2

∫ L/2

−L/2dz

∫ z

−L/2dz′H (z)H (z′)|vac〉 + · · · . (3)

After some algebra, we find to the second order in perturbationthat

|〉 = N |vac〉 − iκL

∫dνsdνi�

(1)(νs,νi)|νs,νi〉

+ i(κL)2∫

dνsdνidν ′sdν ′

i

×�(2)(νs,ν′s ,νi,ν

′i)|νs,ν

′s ,νi,ν

′i〉, (4)

where the coefficient N is determined self-consistently suchthat |〉 is normalized. The states |νs,νi〉 ≡ a

†s (νs)a

†i (νi)|vac〉

and |νs,ν′s ,νi,ν

′i〉 ≡ a

†s (νs)a

†s (ν ′

s)a†i (νi)a

†i (ν ′

i)|vac〉 represent thebases containing single and double pairs of photons, respec-tively. �(1)(νs,νi) and �(2)(νs,νi) are the joint two-photon

(single-pair) and four-photon (double-pair) spectral functions,respectively, defined as

�(1)(νs,νi) = φ(νs + νi)sinc (μsνs + μiνi) , (5)

�(2)(νs,ν′s ,νi,ν

′i) = φ(νs + νi)φ(ν ′

s + ν ′i)

2(μsν ′s + μiν

′i)

× [sinc[μs(νs + ν ′s) + μi(νi + ν ′

i)]

− e−i(μsν′s+μiν

′i )sinc (μsνs + μiνi) ],

(6)

where μs,i = βs,iL/2 are phase-matching coefficients. FromEq. (4), the probability to generate photon pairs is

P = (κL)2p + (κL)4p2 (7)

with

p =∫

dνsdνi |�(1)(νs,νi)|2. (8)

We next model the heralding stage. Existing studies havebeen based on the Schmidt decomposition analysis, in whichthe joint two-photon spectral function (5) is expanded ontoa set of orthogonal Schmidt modes. The purity of heraldedsingle-photon states is then estimated from the expansioncoefficients [15,25,26,29]. This analysis, however, is inap-plicable to photon-pair states containing more than one pairof photons. Moreover, it is not physically rigorous becausesuch Schmidt modes in general are not the eigenmodes ofthe measurement apparatus for the signal photons. Instead, arigorous analysis must be developed following the quantummeasurement postulate, where a detection event collapses thesignal-photon state onto an eigenstate of the measurementapparatus. For the present setup, such eigenstates correspondto a set of time- and bandwidth-limited modes that are chosenby following the standard procedure for the detection ofband-limited signals over the measurement time [37,38]. Inthe frequency domain, the measurement eigenstates containingone and two photons are given by [30,39]

|1〉m =∫ B/2

−B/2dνsϕm(c,νs)|νs〉, (9)

|2〉m = 1√2

∫ B/2

−B/2dνs

∫ B/2

−B/2dν ′

sϕm(c,νs)ϕm(c,ν ′s)|νs,ν

′s〉,(10)

where m = 0,1, . . . is the order number of the eigenstates,and c = πBT/2 is a dimensionless parameter determining themode structure. The mode function is

ϕm(c,νs) =√

(2m + 1)

BS0m

(c,

2νs

B

), (11)

where {Snm(x,y)} are the angular prolate spheroidal functions.Its corresponding eigenvalue is

χm(c) = 2c

π

[R

(1)0m(c,1)

]2 � 1, (12)

where R(1)nm(c,x) is the radial prolate spheroidal function.

Ordering χ0(c) > χ1(c) > χ2(c) · · ·, m = 0 represents thefundamental detection mode of our interest.

033844-2

OPTIMIZED HERALDING SCHEMES FOR SINGLE PHOTONS PHYSICAL REVIEW A 84, 033844 (2011)

With the eigenstates (9) and (10), the positive operator-valued measure (POVM) for registering signal photons by anon-off detector is given by

Pon =∑m

(ηm|1〉mm〈1| + (

2ηm − η2m

)|2〉mm〈2|)+

∑m<m′

(ηm + ηm′ − ηmηm′)|1〉mm〈1| ⊗ |1〉m′m′ 〈1|, (13)

where ηm = ηχm(c) and η is the total detection efficiencyincluding transmission losses and the inherent quantumefficiency of the detector. In arriving at this POVM, wehave assumed that at most two photons reach the detectorsimultaneously, consistent with our previous approximationthat at most two pairs of photons can be created per pumppulse. Furthermore, we have neglected the effects of both darkcounts and afterpulsing in the detector.

With Pon in Eq. (13), the probability of a detector clickis computed via Ps = Tr{Pon|〉〈|}s,i , where the trace iscarried over both the signal and idler photon states. FromEqs. (4)–(6) and (9)–(13), we obtain

Ps = (κL)2P (1)s + (κL)4P (2)

s , (14)

where

P (1)s =

∑m

ηm

∫dνi |ψm(νi)|2, (15)

P (2)s =

∫dνidν ′

i

( ∑m<m′

(ηm + ηm′ − ηmηm′)

× |ψm,m′ (νi,ν′i)|2 +

∑m

(ηm − η2

m

/2)|ψm,m(νi,ν

′i)|2

).

(16)

Here, ψm(νi) and ψm,m′ (νi,ν′i) are the (un-normalized) her-

alded one-photon and two-photon wave functions for the idlerphotons, respectively, defined as

ψ (1)m (νi) =

∫ B/2

−B/2dνsϕm(c,νs)�

(1)(νs,νi), (17)

ψ(2)m,m′ (νi,ν

′i) =

∫ B/2

−B/2dνs

∫ B/2

−B/2dν ′

s�(2)(νs,ν

′s ,νi,ν

′i)

× (ϕm(c,νs)ϕm′(c,ν ′s) + ϕm(c,ν ′

s)ϕm′(c,νs)).

(18)

From Eqs. (7) and (14), the detection efficiency of signalphotons (i.e., the probability for the signal-photon detector toclick given that at least one photon pair is emitted) is given by

Ds = Ps

P. (19)

After a detector click, one or more idler photons areheralded, whose reduced density matrix is given by ρi =Tr{Pon|〉〈|}s/Ps , where the trace is carried over the signal-photon states only. After some algebra, we obtain

ρi = (κL)2P (1)s

Ps

ρ(1)i + (κL)4P (2)

s

Ps

ρ(2)i , (20)

where ρ(1)i and ρ

(2)i are the normalized reduced-density

matrices describing the idler-photon states containing one and

two photons, respectively. They satisfy Tr{ρ(1),(2)}i = 1. Forthe one-photon density matrix, we explicitly obtain

ρ(1)i = 1

P (1)s

∑m

ηm

∫dνidν ′

iψm(νi)ψ∗m(ν ′

i)|νi〉〈ν ′i |. (21)

In order to characterize the properties of the heraldedphotons, we expand ρ

(1)i onto a set of eigenstates {|n〉i},

where ρ(1)i = ∑∞

n=0 λn|n〉ii〈n|, with ordered eigenvalues {λi}such that λ0 > λ1 > λ2 · · ·. We then compute the heraldingefficiency H defined as the probability of finding the idlerphoton in the single-photon state |0〉i upon a detector click.From Eq. (20), we obtain

H = P (1)s

P (1)s + (κL)2P (2)

s

λ0. (22)

We note that there is an alternate definition of heraldingefficiency, which is the probability that an idler photon canbe detected after heralding. That definition, however, doesnot capture the quality of the heralded photons, and somemeasurement giving the purity of the heralded state must bespecified. In contrast, the metricH defined in Eq. (22) capturesboth the detection efficiency and the purity of the heraldedsingle photons.

The other important characteristic is the maximally achiev-able production rate R of the heralded photons, which can bedefined as

R = (κL)2P (1)s + (κL)4P (2)

s

Tmin, (23)

where Tmin is the minimum amount of time required for asingle heralding cycle. In practice, it is given by the largest ofthe following: the detection window T , the temporal lengthof the pump pulses, the temporal length of the signal photonsafter the filter, and the underlying pulse length of the heraldedstate |0〉i . For practical considerations, in this paper we defineexplicitly

Tmin = max[T ,4τp,4τs,4τ0], (24)

where τp, τs , and τ0 are the coherence times of the pumppulses, the filtered signal photons, and the heralded state |0〉i ,respectively (see Ref. [35] for an explicit definition of thecoherence time).

From Eqs. (22) and (23), it is clear that there is an inherenttrade-off between the heralding efficiency H and the single-photon production rate R. Indeed, for given pump bandwidthσ , phase-matching coefficients μs,i , filtering bandwidth B, andmeasurement window T , the values of P (1)

s , P (2)s , and Tmin are

fixed. Then, R increases monotonically with κL. In contrast,H decreases monotonically with κL, due to the growing back-ground emission of multiple photon pairs. Hence, one mustsacrifice the production rateR for a higher heralding efficiencyH, or vice versa. In particular, in the weak pumping regimewherein such single-photon sources operate, the probability ofgenerating multiple photon pairs is much smaller than that ofgenerating a single pair. Thus, to a good approximation,

H =(

1 − (κL)2P (2)s

P (1)s

)λ0, (25)

R = (κL)2 P (1)s

Tmin. (26)

Hence, the trade-off between H and R is approximately linear.

033844-3

YU-PING HUANG, JOSEPH B. ALTEPETER, AND PREM KUMAR PHYSICAL REVIEW A 84, 033844 (2011)

III. OPTIMIZATION OF HERALDING

Using the multimode theory developed in the previoussection, we now study the optimization of measurementschemes for the best heralding performance in the presenceof various photon-pair spectral correlations. This involvesnumerically identifying the appropriate spectral and temporalfiltering windows that simultaneously maximize the heraldingefficiency H and the production rate R.

We first consider the photon-pair state (4) with μs = μi =0. This state exists approximately in a variety of photon-pairsources where the bandwidths of the phase-matching spectraare much broader than those of the pump and the signalspectral filter [40,41]. The two-photon (single-pair) state in thiscase is not factorable according to the Schmidt decompositionanalysis, with the degree of factorization K → 0 in the limitof large B [42]. In Fig. 1 we plot the resulting heraldingefficiency H versus the production rate R for various B.For the sake of comparison, for each B we choose T tomaximizeRwhile fulfillingH � 0.99 in the weak-pump limit.Such chosen T ’s turn out to give an overall optimal trade-offbehavior for 0.9 � H � 0.99, as we show later in this section.As shown in Fig. 1, for each B, H decreases monotonicallywith R, exhibiting the trade-off effect. The optimal trade-off,represented by the topmost H-R curve in the figure, isachieved with B = 0.95σ . For larger or smaller B’s, the H-Rcurves fall below the optimum, showing reduced heraldingperformance.

For comparison, in Fig. 1 we have also plotted theresult obtained for a factorable two-photon state withoutapplying any spectral or temporal filtering [15,25,26,29].We have chosen μs = 25/σ and μi = 0 such that in theweak-pump limit the heralding efficiency H = 0.99. Thesephase-matching parameters correspond to the signal photontraveling at a much slower group velocity than the pumppulse, while the idler photon travels at the pump-pulse velocity.In the joint two-photon spectrum, the underlying two-photon

FIG. 1. (Color online) H as a function of R for various B,shown for a nonfactorable state with μs = μi = 0. The bottom linerepresents the result when a highly factorable state with μs = 25/σ

and μi = 0 is used but neither spectral nor temporal filtering isapplied.

FIG. 2. (Color online) Similar to Fig. 1, but shown for a factorablephoton-pair state with μs = 10/σ and μi = 0.

state corresponds to an ellipse squeezed along the axis ofthe signal-photon frequency [28]. As shown, the heraldingperformance in this case is lower than is achievable withthe nonfactorable state by applying spectral and temporaryfiltering. This result suggests that the use of factorable statesalone does not lead to an optimized trade-off between Hand R.

For the second example, we analyze the heralding per-formance using the photon-pair state (4) with μs = 10/σ

and μi = 0. The two-photon state in this case, similar tothat described in the previous paragraph, corresponds to anellipse in the joint two-photon spectrum. It is approximatelyfactorable according to the Schmidt decomposition, with thedegree of factorability K = 1.03 [42]. In Fig. 2, we plotH versus R for different B’s, where, again, for each B

we optimize T to maximize R while fulfilling H � 0.99in the weak-pump limit. As shown, a qualitatively similartrade-off behavior is found between H and R as in Fig. 1. Theoptimal heralding performance, represented by the topmostH-R curve, is achieved with B = 0.6σ . For smaller B’s, thetrade-off degrades rapidly. Note that for B = 0.2σ , the H-Rcurve falls below the reference curve obtained for a highlyfactorable state with μs = 25/σ , μi = 0, and no filtering. Thissuggests that for factorable states, too much spectral filteringlowers the heralding performance. For larger B’s, on theother hand, the trade-off behavior degrades only slowly. Note,however, if no filtering is applied, the heralding performancewill be quite low, as discussed in the previous paragraph.

Comparing Figs. 1 and 2, we see that for a similarfilter bandwidth B, the trade-off behavior between H andR is different for different photon-pair correlations. Byapplying the optimized measurement for each correlation,however, the best trade-off behavior that can be achieved turnsout to be quite similar, as evidenced by the similarity betweenthe topmost curves in Figs. 1 and 2. This result suggests thatthere is no inherent relation between the achievable heraldingperformance and the spectral-correlation properties of thephoton pairs used for heralding. To show this clearly, inFig. 3(a) we plot the production rate R0 as a function of B

033844-4

OPTIMIZED HERALDING SCHEMES FOR SINGLE PHOTONS PHYSICAL REVIEW A 84, 033844 (2011)

FIG. 3. (Color online) (a) R0 as a function of B. (b) Ropt0 versus

the photon-pair purity Pr .

for different pair correlations, with the pump power chosensuch that the heralding efficiency H = 0.95 in each case.As shown, for μs = μi = 0, R0 is peaked at B = 0.95σ ,for which the optimal rate Ropt

0 = 0.0013σ is achieved. Toobtain this optimum, the time window is set at T = 1.1/σ .The single-pair generation rate is 5.7% per pump pulse (thedouble-pair rate is 0.32%) and the heralding repetition rateis 1/Tmin = σ/2.7. The detection probability Ds , assuming atotal detection efficiency of η = 10%, is 5.9%. For B < 0.95σ ,the system enters the narrowband spectral filtering regimewhere R0 decreases rapidly with B. In the opposite broadbandfiltering regime where B > 0.95σ , T must be quite small inorder to achieve a high heralding efficiency [30]. In effect,the system is then operated in the tight temporal-filteringregime [43] with R0 decreasing for increasing B. Hence, fornonfactorable states, very tight spectral or temporal filteringwill lead to poor trade-off behavior between the heraldingefficiency and the production rate. Only for an appropriatecombination of moderate spectral and temporal filtering canboth high heralding efficiency and high production rate besimultaneously achieved.

In Fig. 3(a), for the factorable state with μs = 10/σ

and μi = 0, the optimal production rate Ropt0 = 0.0016σ is

achieved with B = 0.6σ . At this optimum, the single-pairemission rate is 8.9% per pump pulse (the double-pair rate is0.8%) and the heralding repetition rate is σ/3.2. The detectionwindow T = 3.2/σ and the detection efficiency Ds = 5.3%for η = 10%. For B < 0.6σ , R0 decreases rapidly with B.On the other hand, for B > 0.6σ , R0 decreases relativelyslowly when B increases. This behavior shows that a precisecontrol of spectral filtering is not required for factorable

photon-pair states. In comparison, for a less factorable statewith μs = 2.6/σ and μi = 0, a similar behavior is shownin Fig. 3(a) but with Ropt

0 = 0.0014σ that is achieved atB = 0.85σ .

Also in Fig. 3(a), we have plotted the results for anothertype of factorable photon-pair states, which, instead of ellipses,correspond to circles in the joint two-photon spectrum [25]. Weconsider two states of this type. The first is with μs = −1.33/σ

and μi = 0.45/σ , which could be created in potassium-titanyl-phosphate waveguides [27]. As shown, the optimal Ropt

0 =0.0014σ , which is achieved with B = 0.85σ . The second statecorresponds to μs = −1.3/σ and μi = 1.3/σ , which couldbe generated in photonic-crystal fibers [26]. As shown, theoptimal Ropt

0 = 0.0012σ , achieved with B = 0.7σ .As seen in Fig. 3(a), while the optimization conditions vary

significantly with the type of spectral correlations present in thephoton pairs, the optimized single-photon production rates aremore or less the same regardless of the correlation properties.This result suggests that by adopting optimized filtering, anyspectral correlation can lead to a similarly high heraldingperformance. In other words, increasing the factorability ofthe underlying photon-pair states would not (significantly)improve the heralding performance. To clearly show this, inFig. 3(b) we plot Ropt

0 versus the purity Pr = 1/K of thetwo-photon (single-pair) states calculated via the Schmidtdecomposition [42]. Each of the data points (1)–(5) correspondto the photon-pair states (1)–(5) listed in Fig. 3(a). Point (6)is the result for the highly factorable photon-pair state withμs = 25/σ and μi = 0, without applying any spectral ortemporal filtering. As shown, despite a large variance in thefactorability of the photon-pair states, ranging from Pr = 0to Pr ≈ 1, the maximum production rates remain nearly thesame when the optimized measurement is employed for eachcorrelation. For a highly factorable state but without filtering,in contrast,Ropt

0 is much smaller, as shown by the data point (6)in Fig. 3(b).

Thus far, we have studied the optimization of measurementsfor heralding by surveying B. For each B, we have chosenT to maximize R while maintaining H � 0.99 in the weak-pump limit. Such B and T , strictly speaking, may not give themaximum R for every H, or vice versa. In practice, however,the figure of merit for a single-photon source is the trade-off relation between R and H, which ultimately limits theperformance of a single-photon-based application. A well-known example is the BB84 quantum cryptography, where theobtainable fresh-key generation rate is limited fundamentallyby the trade-off between the quantum-bit error rate, which isdetermined byH, and the raw-key rate, which is determined byR [3]. A goal of this paper is to study how to optimally mitigatethe R-H trade-off in order to achieve the best performance insuch applications.

To show that the optimal trade-off behavior is indeedachieved via our optimization approach, in Fig. 4 we plotH versus R for various choices of B and T , with μs = μi = 0as considered in Fig. 1. For B = 0.95σ and T = 0.7/σ ,H approaches 0.995 in the limit of small R. However, itdecays rapidly as R increases. Hence, a small T is superiorfor pair generation at very high heralding efficiencies butlow production rates. For B = 0.95σ and T = 2.3/σ , in

033844-5

YU-PING HUANG, JOSEPH B. ALTEPETER, AND PREM KUMAR PHYSICAL REVIEW A 84, 033844 (2011)

FIG. 4. (Color online) The heralding efficiency H as a functionof the single-photon production rate R for μs = μi = 0 and a varietyof B’s and T ’s.

contrast, H is 0.97 in the small R limit. Yet, it decays moreslowly with R. A large T is thus advantageous for pairgeneration at relatively lower heralding efficiencies but higherproduction rates. A moderate T , therefore, will give rise toa more balanced trade-off behavior in the two regimes. ForT = 1.1/σ , in particular, the H-R curve is close to optimal inboth regimes, exhibiting good overall heralding performance.In Fig. 4 we also plot the results for a variety of B’s and

T ’s. Comparing all the plotted curves confirms that our choiceof B = 0.95σ and T = 1.1/σ indeed globally optimizes theheralding performance for 0.9 � H < 0.99. Lastly, we havesimilarly validated our optimization method for the other typesof photon-pair correlations considered above.

IV. CONCLUSION

We have developed a multimode theory for the descriptionof heralded generation of single photons using waveguide orfiber-based photon-pair sources. Our theory takes into accountthe background emission of multiple photon pairs duringeach pump pulse, as well as the multimode nature of thetime-bandwidth-limited measurements of photonic signals.Based on this theory, we have numerically identified theoptimized measurement schemes that give rise to the bestheralding performance in the presence of various photon-paircorrelations. Interestingly, we have discovered that with theoptimized measurement, similar heralding performance can beachieved irrespective of the spectral-correlation property of theused photon-pair source. This suggests that instead of tailoringthe photon-pair sources for prescribed correlation properties,the key to improving the heralded generation of single photonsis to appropriately measure, via spectral and temporal filtering,the signal photons that are detected for heralding.

ACKNOWLEDGMENTS

This research was supported in part by the DefenseAdvanced Research Projects Agency under Grant No.W31P4Q-09-1-0014 and by the United States Air Force Officeof Scientific Research under Grant No. FA9550-09-1-0593.

[1] D. Bouwmester, A. Ekert, and A. Zeilinger, The Physicsof Quantum Information: Quantum Cryptography, QuantumTeleportation, Quantum Computation (Springer, Berlin, 2000).

[2] E. Knill, R. Laflamme, and G. J. Milburn, Nature (London) 409,46 (2001).

[3] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod.Phys. 74, 145 (2002).

[4] N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. deRiedmatten, and N. Gisin, Phys. Rev. A 76, 050301(R) (2007).

[5] H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39,691 (1977).

[6] F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987).[7] F. De Martini, G. Di Giuseppe, and M. Marrocco, Phys. Rev.

Lett. 76, 900 (1996).[8] J. Kim, O. Benson, H. Kan, and Y. Yamamoto, Nature (London)

397, 500 (1999).[9] P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F.

Strouse, and S. K. Buratto, Nature (London) 406, 968 (2000).[10] C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986).[11] P. Grangier, G. Roger, and A. Aspect, Europhys. Lett. 1, 173

(1986).[12] A. V. Sergienko, M. Atature, Z. Walton, G. Jaeger, B. E. A.

Saleh, and M. C. Teich, Phys. Rev. A 60, R2622 (1999).

[13] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek,and S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).

[14] J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I.Vistnes, and E. S. Polzik, Opt. Express 15, 7940 (2007).

[15] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B.U’Ren, C. Silberhorn, and I. A. Walmsley, Phys. Rev. Lett. 100,133601 (2008).

[16] J. Fulconis, O. Alibart, W. Wadsworth, P. Russell, and J. Rarity,Opt. Express 13, 7572 (2005).

[17] E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, andA. Migdall, Phys. Rev. A 78, 013844 (2008).

[18] O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley,and I. A. Walmsley, Phys. Rev. Lett. 102, 123603 (2009).

[19] A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity,and W. J. Wadsworth, Opt. Express 17, 6156 (2009).

[20] J. Chen, K. F. Lee, C. Liang, and P. Kumar, Opt. Lett. 31, 2798(2006).

[21] A. Zeilinger, M. A. Horne, H. Weinfurter, and M. Zukowski,Phys. Rev. Lett. 78, 3031 (1997).

[22] M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, Photon.Technol. Lett. 27, 491 (2002).

[23] R. Kaltenbaek, B. Blauensteiner, M. Zukowski, M. Aspelmeyer,and A. Zeilinger, Phys. Rev. Lett. 96, 240502 (2006).

033844-6

OPTIMIZED HERALDING SCHEMES FOR SINGLE PHOTONS PHYSICAL REVIEW A 84, 033844 (2011)

[24] J. Fulconis, O. Alibart, J. L. O’Brien, W. J. Wadsworth, andJ. G. Rarity, Phys. Rev. Lett. 99, 120501 (2007).

[25] W. P. Grice, A. B. U’Ren, and I. A. Walmsley, Phys. Rev. A 64,063815 (2001).

[26] K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S.Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J.McKinstrie, S. Radic, and I. A. Walmsley, Opt. Express 15,14870 (2007).

[27] Z. H. Levine, J. Fan, J. Chen, A. Ling, and A. Migdall, Opt.Express 18, 3708 (2010).

[28] M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong,W. J. Wadsworth, and J. G. Rarity, Opt. Express 17, 4670(2009).

[29] C. Soller, B. Brecht, P. J. Mosley, L. Y. Zang, A. Podlipensky,N. Y. Joly, P. S. J. Russell, and C. Silberhorn, Phys. Rev. A 81,031801 (2010).

[30] Y.-P. Huang, J. B. Altepeter, and P. Kumar, Phys. Rev. A 82,043826 (2010).

[31] The present model does not include the effect of Ramanscattering, which is usually not negligible for existing fiber-based photon-pair sources. Such an effect can neverthelessbe mitigated in various ways, including lowering the fibertemperature.

[32] Instead of an on-off detector, one can use a photon-number-resolving detector, by which the effect of multiple photon-

pair emission can be suppressed, providing a near-unity totaldetection efficiency.

[33] M. A. Hall, J. B. Altepeter, and P. Kumar, in Conference onLasers and Electro-Optics (CLEO) and the Quantum Electronicsand Laser Science Conference (QELS) (Optical Society ofAmerica, Washington, DC, 2010), paper QThG6.

[34] M. A. Hall, J. B. Altepeter, and P. Kumar, Phys. Rev. Lett. 106,053901 (2011).

[35] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics(Cambridge University Press, Cambridge, 1995).

[36] Q. Lin, F. Yaman, and G. P. Agrawal, Phys. Rev. A 75, 023803(2007).

[37] D. Splepian and H. O. Pollak, Bell Syst. Tech. J. 40, 43(1961).

[38] C. Zhu and C. M. Caves, Phys. Rev. A 42, 6794 (1990).[39] M. Sasaki and S. Suzuki, Phys. Rev. A 73, 043807

(2006).[40] M. A. Hall, J. B. Altepeter, and P. Kumar, Opt. Express 17,

14558 (2009).[41] M. Medic, J. B. Altepeter, M. A. Hall, M. Patel, and P. Kumar,

Opt. Lett. 35, 802 (2010).[42] P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley,

New J. Phys. 10, 093011 (2008).[43] M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and

H. Zbinden, Nature Phys. 3, 692 (2007).

033844-7